The proof follows by uniqueness of \(U_\beta ^j\)  and translation invariance in \(x\)  of (REF ), see Proposition 2.10 (b) in [1]}. It is more convenient to work with solutions of (REF ) since they are defined for all spatial points in \(\mathbb {R}^N\) . Finally, any result will then be inherited by \(\widetilde{\overline{U}}\) . To ease the notation, \(U^j\)  will denote solutions of (REF ) and its time interpolant as \(\widetilde{U}\) .
